Application bilinéaire

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Let and be three vector spaces over the same base field . A bilinear map is a function such that for all , the map is a linear map from to and for all , the map is a linear map from to In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map satisfies the following properties. For any , The map is additive in both components: if and then and If and we have B(v, w) = B(w, v) for all then we say that B is symmetric. If X is the base field F, then the map is called a bilinear form, which are well-studied (for example: scalar product, inner product, and quadratic form). The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It generalizes to n-ary functions, where the proper term is multilinear. For non-commutative rings R and S, a left R-module M and a right S-module N, a bilinear map is a map B : M × N → T with T an (R, S)-bimodule, and for which any n in N, m ↦ B(m, n) is an R-module homomorphism, and for any m in M, n ↦ B(m, n) is an S-module homomorphism. This satisfies B(r ⋅ m, n) = r ⋅ B(m, n) B(m, n ⋅ s) = B(m, n) ⋅ s for all m in M, n in N, r in R and s in S, as well as B being additive in each argument. An immediate consequence of the definition is that B(v, w) = 0X whenever v = 0V or w = 0W. This may be seen by writing the zero vector 0V as 0 ⋅ 0V (and similarly for 0W) and moving the scalar 0 "outside", in front of B, by linearity. The set L(V, W; X) of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from V × W into X. If V, W, X are finite-dimensional, then so is L(V, W; X).